Asymptotic Analysis of von Neumann Entropy in Conformal Field Theory

Given a QFT net A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {A}}}$$\end{document} of local von Neumann algebras A(O)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {A}}}(O)$$\end{document}, we consider the von Neumann entropy SA(O,O~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{{\mathcal {A}}}(O, {\widetilde{O}})$$\end{document} of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras A(O)⊂A(O~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {A}}}(O)\subset {{\mathcal {A}}}({\widetilde{O}})$$\end{document} (split property). This canonical entanglement entropy SA(O,O~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{{\mathcal {A}}}(O, {\widetilde{O}})$$\end{document} is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). The finiteness property is derived by an explicit formula of entropy and an observation that the operators in the definition are closely related to Hankel operators. In this paper we give further analysis of this entropy using a variety of techniques that have been developed in different context, and in particular we show that there is an upper bound given by a positive constant multiply by |lnη|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ln \eta |$$\end{document}, where η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} is the cross ratio of the underlying system, when η→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \rightarrow 0$$\end{document}.


Introduction
von Neumann entropy is the basic concept in quantum information and extends the classical Shannon's information entropy notion to the non commutative setting. The role of entanglement in Quantum Field Theory is more recent and increasingly important; it represents a piece of the quantum information framework in this subject. It appears in relation with several primary research topics in theoretical physics as area theorems, c-theorems, quantum null energy inequality, etc. (see for instance [5,6,40] and refs. therein).
Despite the rich physical literature on the subject, the rigorous definition of entanglement entropy in QFT is however not obvious. The point is that the von Neumann algebra

von Neumann algebra A(O) ∨ A(O ) generated by A(O) and A(O ) is equal to B(H), a type I factor, and cannot be naturally isomorphic to the von Neumann tensor product A(O) ⊗ A(O ) which is type I I I .
To get rid of short distance divergences, one may however consider a slightly larger double cone O ⊂ O, namely the closure of O is contained in the interior of O. The split property states that there is a natural isomorphism of von Neumann algebras The split property expresses the statistical independence of A(O) and A( O ); it was verified for the free, neutral Boson QFT case in [2]. It was studied in [11] and led to important structural features both in Mathematics and in Physics. It follows under natural, general physical requirements [4]. It holds automatically in chiral conformal QFT [28]. (See [16] for a discussion of its validity in topologically non trivial spacetimes).
The split property is a local property, in fact it is equivalent to the existence of an intermediate type I factor F between A(O) and A( O) A type I factor F is a von Neumann algebra isomorphic to B(K), the algebra of all bounded linear operators on some Hilbert space K. This definition however depends on the choice of F. Actually, if the split property holds, there are infinitely many intermediate type I factors F in (1). Yet, as shown in [11], there is a canonical intermediate type I factor F, associated with the O, O and the vacuum vector , given by the formula (if the local von Neumann algebras are factors), with J is the modular conjugation of the relative commutant von Neumann algebra A(O) ∩ A( O) associated with . We then define the (canonical) entanglement entropy of A with respect to O, O as where F is the canonical intermediate type I factor (2). Here Tr is the trace of F (namely F = B(H A ) ⊗ 1 H B and tr corresponds to the usual trace on B(H A )) and ρ F is the vacuum density matrix relative to F. The above definition concerns a local net A. If A if a Fermi net, graded locality rather than locality holds. In this case, the split property is still defined by (1) and the entanglement entropy by (3). However, the canonical intermediate type I factor is to be defined by a twisted version of formula (2), cf. equation 50 of [25].
A main result in [25] is that above defined canonical entanglement entropy is finite for the chiral conformal net M generated by a complex free fermion on S 1 . Here, double cones are intervals I ⊂ I of S 1 .
In fact in [25] an explicit formula for the von Neumann entropy is given, and its finiteness follows from observing the connection to the theory of Hankel operators . It is in fact the first known case where such canonical entropy is proved to be finite. It is therefore a natural question to estimate this finite entropy, in particular its asymptotic property as the cross ratio (sin(η/2)) 2 goes to zero or equivalently when the end points of the interval get close to each other (cf. Remark 3.9 in [25]) . Note that due to the monoticity of relative entropy the entropy is bounded below (cf. Lemma 2.4) by 1 6 | ln sin(η/2)|, so the real interest is about its upper bound. Our result (cf. Cor. 3.17 ) is that the upper bound is again a constant multiplied by | ln η| as η → 0. The proof of this result is surprisingly delicate and rely on deep results in [37], [21] and [38]. The results of [37], [21] and [38] are motivated by questions of semi-classical analysis of entropy in QFT, and the context of these questions are very different from ours. In fact since our functions are not smooth on the circle, we have to modify the proof of some of the results in these papers for our analysis. These modifications include Lemma 3.11 which is based on a result in [38], but now applied in three different scales in the proof of Th. 3.15. By using properties of Hankel operators, it turns out that we can do our estimates by removing the poles of our functions inside the unit disk. We then use a change of part of the path to evaluate the fourier coefficients of our functions, first in the relatively easy case when our functions have no branch cuts. When our functions have branch cuts inside the unit disk, we reduce our analysis to the estimation of Besov quasinorm of these functions (cf. Section 3.3). We expect that our techniques will have applications in more general cases.
There are some similarities between our entropy and reflective entropy discussed in the physics literature (cf. [9] and references therein). In [9] there are also numerical computations of such reflective entropy and their numerical data agrees with our asymptotic analysis, but it is not clear at all that those numerical computations on finite lattices in [9] actually converge to our entropy. It is an interesting question to further understand this similarity. It is also an interesting question to improve our estimates in this paper, in particular to determine the constant in Cor. 3.17. See Remark (3.18) and Remark (3.8).
The rest of this paper is organized as follows. In Section 2 we recall the entropy defined in [25] in the context of chiral net of free fermion, and recall some basic facts related to Hankel operators in [30]. We begin our asymptotic analysis in Section 3. Our basic idea is explained at the beginning of Section 3.1. Roughly speaking we deform the path of integration, removing the poles of our functions inside the unit disk, and then estimate Besov quasi-norms of these functions (cf. the proof of Th. 3.7). We then use these results in Section 3.4 to give the Schatten norm of our functions. This is based on Lemma 3.11, and a key result in Th. 3.15. In Section 3.5 we prove Cor. 3.17 as a consequence of our results in the previous two sections and results of [21]. In the last Section we show that our entropy function is continuous in η and goes to 0 as η goes to π .

Schatten-von Neumann
Ideals. This paper relies on the results for general quasinormed ideals of compact operators. Here we limit our attention to the case of Schattenvon Neumann operator ideals S q , q > 0. Detailed information on these ideals can be found e.g. in [30] and [36]. We shall point out only some basic facts. For a compact operator A on a separable Hilbert space H , denote by s n (A), n = 1, 2, ... its n-th singular values, that is, the eigenvalues of the operator |A| := √ A * A. Note that if R 1 , R 2 are bounded operators, Then (cf. [36]) where ||A|| to denote the norm of an operator A. We denote the identity operator on H by 1. The Schatten-von Neumann ideal S q , q > 0 consists of all compact operators A , Note that |A| S q = |A * | S q . If q ≥ 1 , then the above functional defines a norm; if 0 < q < 1 , then it is a so-called quasi-norm. There is nevertheless a convenient analogue of the triangle inequality, which is called the q-triangle inequality: We also have the Holder inequality: See [19] and also [3]. In what follows we focus on the case q ∈ (0, 1]. We will use ||A|| to denote the norm of an operator, and ||A|| 1 the trace of |A|. By definition Note that for a nonzero operator A, the singular values of A/||A|| is bounded above by 1, therefore if 0 < p < q ≤ 1 we have |A/||A|||  [31]. For more details, see [31] or [39]. Such a representation gives rise to a graded net as follows. Denote by A(I ) the von Neumann algebra generated by c(ξ ) s, with ξ ∈ L 2 (I, C). Here c(ξ ) = a(ξ ) + a(ξ ) * and a(ξ ) is the creation operator defined as in Chapter 1 of [39]. Let Z : F p → F p be the Klein transformation given by multiplication by 1 on even forms and by i on odd forms. It For bounded operators A, B : where is an operator on F p given by multiplication by 1 on even forms and −1 on odd forms. Note that Z = 1−i 1−i . An operator A is called even (resp. odd) if A = A + (resp. A = A − ). Note that ω(a) = 0 if a is odd, where ω is the vacuum state corresponding to the vacuum vector . We set By (1) Lemma 3.1 in [24] this defines a normal state on the von Neumann algebra generated by A(I 1 ) and A(I 2 ). The mutual information S(ω, ω 1 ⊗ 2 ω 2 ) (cf. Definition 2.1 of [24]) is computed in Section 3 of [24].

von Neumann Entropy from Split Property.
In this section we recall the von Neumann entropy defined in [25] from split property that we aim to compute in this paper.

General Symmetric Interval
We will focus on the one particle structure on L 2 (S 1 ; C) in this section. On S 1 , we consider the following general four "symmetric intervals" We will denote by η = π − φ. See Figure 1. Denote by I 0 := {e iθ : 0 < θ < 2φ}. For any interval I if we denote by I 2 the set of z such that z = w 2 for some w ∈ I , then it is clear that I 0 = I 2 1 . We shall consider the action of SU (1, 1) on S 1 which is given by z → az+b bz+ā with |a| 2 −|b| 2 = ±1. The Möbius group Mob is the subgroup of SU (1, 1) of elements with determinant |a| 2 − |b| 2 = 1. The action z → 1 z is orientation reversing. This element has a = d = 0, b = c = −1. If m(z) = az+b bz+ā , the unitary action of m on S 1 is given by (See Section 4 of [39]) Since (a −bz) −1 is holomorphic for |z| < 1 and |a| > |b|, U m and its inverse preserves P H, and so U m commutes with the Hardy space projection P. The flip map ( By sending the orientation reversing element z → 1 z in SU (1, 1) to F 1 , we get an action of SU (1, 1) on H which is of the form where m(z) = az+b bz+ā , α m (z) = (a −bz) −1 . Let m ∈ Mob be such that m I 0 is the upper half circle. Let m 1 = m −1 F 1 m. It is straightforward to see that Define We have the following definition where Note that m 1 (z 2 ) depends on η. Note that j maps L 2 (I 1 ) to L 2 (I 2 ∪ −I 2 ). We will denote by M I the multiplication operator by χ I , the characteristic function of interval I . By definition, M I 1 and j M I 1 j are orthogonal projections whose ranges L 2 (I 1 ) and j L 2 (I 1 ) are also orthogonal. Hence P 12 := M I 1 + j M I 1 j is a projection. If we wish to emphasize the dependence on η we will write P 12 as P 12 (η).
Note that L 2 (I 1 ) ⊕ j L 2 (I 1 ) is the canonical type I standard subspace which is intermediate between L 2 (I 1 ) and L 2 (I 1 ∪ I 2 ∪ −I 2 ). The following is theorem 3.3 from [25]: Theorem 2.1.
is a projection onto L 2 (I 1 ) ⊕ j L 2 (I 1 ) where We note that since g, h are invariant under u → −u, these functions are independent of the choice of the square root u of m 1 (z 2 ) in Th. 2.1.
For simplicity we will write multiplication operator such as M g simply as g when no confusion arises. For an example we can write P 12 = g + h R. Similarly we write g * the complex conjugate of a function such as g.
The von Neumann entropy S(F, η) that comes from the canonical type I standard subspace We make the following definition: where P 12 is as in Th. 2.1. For any operator self adjoint T and continuous f defined on the spectrum of T and PT P, [25], and we put in extra η to emphasize the dependence on η. Let us first explain why S(F, η) = S(F) where S(F) is as in Th. 3.8 of [25]. Let us first recall how S(F) is defined. We shall denote by F := L 2 (I 1 )⊕ j L 2 (I 1 ) the canonical type I standard subspace. Recall that on H = L 2 (S 1 ; C), the complex structure on H is given by i(2P − 1), with P the projection onto the Hardy space. Let F be the orthogonal complement of i(2P − 1)F. Let P F := P 12 (η), P F be the projections onto F, F respectively. Then P F = (2P − 1)(1 − P F )(2P − 1), and By examing the spectrum of F ( F +1) 2 P F as in Lemma 2.7 in [25] we see that the list (counting multiplicities) of eigenvalues of Since (cf. Chapter 1 of [36]) the set of nonzero elements in the spectrum of T * T is the same as the set of nonzero elements in the spectrum of T T * for a bounded operator T , it follows that the list of nonzero eigenvalues (counting multiplicities) of P P F P = P P 12 (η)P = P P F P F P is the same as that of P F P P P F = P F P P F . So we have S(F, η) = S(F).
In [25] we proved that S(F, η) is finite by observing its connection with Hankel operators. This relies on the growth of fourier coefficients of g, h. We recall the following result which is proved in [25] and follows essentially from an observation of [18], see Th. 3.7 in [25].

P| S q is bounded by a constant which only depends on C and
. Note that this matches with equation (34) of [25]. For the reader who may be confused with the equation (34) of [25], we note that in the definition of h in 2.1 it is important that we have χ I 1 (u), not χ I 1 (z).
If u(z) ∈ I 1 , then g(z) = (u+z) 2 4uz = 1 2 + 1 4 ( u z + z u ). Note that |u| = |z| = 1, it follows that g ≥ 0. Similarly g ≥ 0 if u(z) ∈ −I 1 . When z ∈ I 1 , g = 1, and g = 0 when z ∈ −I 1 . So g ≥ 0. Similarly we can check that ih is real. g − 1/2 and h are both odd functions of z, and g 2 − g = h 2 . To do computations for η close to 0, it is convenient to choose an analytic continuation of u inside the unit disk. Recall that u 2 = m 1 (z 2 ) = z 2 cos η−e −iη e iη z 2 −cos η . Note that the roots of z 2 cos η − e −iη are outside the unit disk. For the square root of z 2 cos η − e −iη , we can choose any branch cut outside the closed unit disk, for an example, two half lines coming out of the two roots of the equation z 2 cos η − e −iη = 0 which do not intersect the closed unit disk. The two roots of e iη z 2 − cos η are inside the disk, and for the square root of e iη z 2 − cos η, we can choose the branch cut to be the closed line segment connecting ±e −iη/2 √ cos η which are the poles of u (cf. Page 128 of [1] for such a choice of branch cut for square root of a quadratic function). u is then the quotient of these two functions. u is an analytic function in the unit disk minus the branch cut. We will see that this branch cut is important for our analysis when η → 0.
On I 2 (e iη z − cos η) 3 (21) Let L(z) := |z 2 − e −iη cos η|. Let us explain how to estimate the derivatives of g, h when η is sufficiently small. We will use g as an example since h is similar. If u(z) ∈ I 1 , then g(z) = (u+z) 2 4uz = 1 2 + 1 4 ( u z + z u ). Hence if u(z) ∈ I 1 the derivatives of g are linear combinations of derivatives of u z and z u . To compute the derivatives of u(z), by definition u(z) 2 = m 1 (z 2 ). So by Chain Rule u(z) u(z) = m 1 (z 2 )z. Keep in mind |u| = |z| = 1. It follows that g is up to addition by a bounded function m 1 (z 2 ) multiplied by a bounded function. g is the sum of a bounded function, ((m 1 (z 2 ))) 2 multiplied by a bounded function and m 1 (z 2 ) multiplied by a bounded function. The same idea applies to all other cases. Note that when η is sufficiently small L(z) ≥ 1/2η 2 , and therefore from (21) we have From this we have where O(1) and C are constants independent of η.
Note that the minimal value of L(z 2 ) is when z = e −iη/2 , i.e. when z is at the middle point of I 2 , and L(e −iη/2 ) = |1 − cos(η)| ∼ 1 2 η 2 when η is close to 0, it follows that on where C is independent of η and η is close to 0. Note that since g = 1 on I 1 , the above formula for g also holds on I 1 .
Note that m 1 (z) is conjugate to the flip, and fix the end points of I 0 . When z is at the end points of I 2 or −I 2 , z 2 takes values at the end points of I 0 . It follows that at the end points of (20) we can see that g is continuous, and g = 0 on the boundary of I 2 , −I 2 . g exists at all points on the circle except the four boundary points of I 2 , −I 2 and is bounded. Since g = 0 on the boundary of I 2 , −I 2 , it follows that the second derivative of g in the distribution sense agrees with g , and in particular it is essentially bounded. Hence g ∈ W 2,∞ . Similarly from formula (20) we see that h is not continuous, but h (z) on I 2 ∪ −I 2 is bounded when z is close to the boundary of I 2 , −I 2 .
The image of I 2 (resp. −I 2 ) under V is interval (1, tan ψ) (resp. (−1, − cot ψ)) with ψ = (π/4 + η/2). The cross ratio of the interval −I 1 , I 1 in the clockwise order is First we recall the lower bound of S(F, η): Denote by ω F , ω F the restriction of the vacuum state to F and its commutant F . Let ω be the tensor state Let us show that when restrictingω to A(I 1 ) ∨ A(−I 1 ), this is the same as ω 1 ⊗ 2 ω 2 as in Sect. 2.2. Since F, F are type I factors and Ad are automorphisms of F (resp. F ) of order two, it follows that = u 1 u 2 where u 1 (resp. u 2 ) is unitary element in F (resp. F ). Multiplying by a phase factor if necessary we can choose u 2 1 = 1, u 2 2 = 1. Since u 1 = u 3 1 = u 1 , it follows that u 1 is an even element, and similarly u 2 is an even element.
By definitionω is the same as ω 1 ⊗ 2 ω 2 on elements of the form ab + , where a ∈ A(I 1 ), and b + is an even element of Note that u 2 Zb − Z −1 is an odd element in F and so ω(u 2 Zb − Z −1 ) = 0. By monotonicity of relative entropy (cf. Chapter 5 of [26]) By Th. 3.16 in [24] we have proved the Lemma.

An Inequality from Besov Quasinorm.
We proceed now to the Besov classes B 1 p p for 0 < p < 1. Let F be an infinitely differentiable function on the real line such that F ≥ 0, with support in [1/2, 2], and n≥0 F( x 2 n ) = 1, ∀x ≥ 1. It is very easy to construct such a function. We can take a nonnegative smooth function F on the interval [1/2, 1] such that F(1/2) = 0, and F(x) = 0, when x is outside [1/2, 2]. Given an analytic function G in the unit disk with continuous extension to the boundary, assume that G(z) − G(0) = n>0 G n z n , define F n * G(z) = j≥1 F( j/n)G j z j where n ≥ 1 is an integer. Note that F n * G(z) is a trignometric polynomial of degree less than 2n. By definition we have G(z) − G(0) = m≥0,n=2 m F n * G. Let z = e 2πit . It follows from Page 250 of [30] This inequality will play a crucial role in our paper. We shall refer to ∞ m=0,n=2 m 2 m −1/2≤t≤1/2 |F n * G(z)| q dt as the Besov quasinorm of G. The definition depends on the choice of F, and all choices give equivalent seminorms, but we shall not make use of this fact. The other direction that It is interesting to note that if t is real, F m (e 2it ) is a periodic function in t and therefore can be thought as a function on the unit circle, but m(F F)(−mt) is not. Nevertheless m(F F)(−mt) captures the dominating part of F m (z) when m → ∞ as the above Lemma shows. By repeatedly using integration by parts we have where s is the imaginary part of t, and C N is a constant which only depends on F.

Asymptotic Analysis
We will determine the upper bound of S(F, η) in the next few sections. Let us first describe the basic ideas. We note that P P 12 P = Pg P + Ph P R , and both Pg P and Ph P (cf. Th. 2.1 for definitions) are Toeplitz operators. When η is small, the support of h shrinks to zero size, so we expect the main contribution to S(F, η) should come from Pg P. To do this we first need to have a good control on the Schattern-von Neumann norm of Ph(1 − P), this is done in Sect. 3.3. There is also a further complication concerning Pg P. It turns out that f 0 (Pg P) is not trace class, but f 0 (Pg P) − P f 0 (g)P is. This problem is addressed in Sect. 3.4. Suppose a function F(z, η), z ∈ S 1 is defined on the circle which depends also on a parameter 0 < η < π. We always assume that F is bounded, i.e., |F(z, η)| ≤ M for some constant M which is independent of z, η. We are interested in the property of F(z, η) when η → 0.  F(z, η) ∼ G(z, η) if there exist two positive constants C 1 , C 2 such that (1) If F is good (resp. very good) , then |F P − P F| q S q = o(− ln η) (resp. O(1)) ; (2) If F and G are good (resp. very good), then both F + G and F G are good (resp. very good).
Proof. By equation (7) we may assume that F, G are good for the same q.
Ad (2): The statement for F + G follows from the q-triangle inequality as in (5). (2) is proved.

Deformation of path.
We'd like to use Lemma 2.3 to do estimation. For this purpose it is important to estimate the growth of the Fourier coefficients of our functions such as h, g. Unfortunately h grows like (cf. equation (22)) η 2 L(z) 3 on I 2 , and L(z) ∼ (θ + η 2 ) where θ is the distance between z and the middle point of I 2 . This makes it difficult or even impossible to obtain O(n −2 ) type estimate. One simple idea is to see if we can use Cauchy's theorem to deform the path I 2 to a path where h is better controlled. A natural such path is the path N which join the ends of I 2 inside the unit disk with property |L(z)| = sin η. On this path N , h ∼ 1 η , and when integrated over N which has length ∼ πη will give us O(1). But we have to pay close attention to possible poles and branch cuts enclosed by I 2 and N . We will see that ultimately it is the branch cut that is responsible for the asymptotic growth of our entropy.

Lemma 3.3. Assume that F(z, η) is analytic in the interior bounded by I 2 and N in the unit disk, and has continuous first derivative on I 2 . In addition assume on the circle F is 0 at the boundary of I 2 , and F is O(1) on the boundary of I 2 . If N |F | = O(1) where N is the path which join the ends of I 2 inside the unit disk with property |L(z)| = sin η.
Then a n (F) := I 2 Fz n dz = O(n −2 ), ∀n ≥ 0.
Proof. By our assumptions on F and integration by parts a n = I 2 Fz n dz = 1 (n + 1)(n + 2) It is sufficient to check that Since F is analytic in the unit disk, deforming the path I 2 to N , and keep in mind |z n | ≤ 1, n ≥ 0 when |z| ≤ 1 we have Proof. By definition

It follows that
Pu 2 (1 − P) = 0 since u 2 is analytic outside the unit disk, and by using Laurent series for u 2 Note that (1 − P)T 1 P is a rank one operator by using Laurent series for T 1 , and the norm of PT 1 (1− P) is given by the maximum of |T 1 | on the circle. These are very special cases of finite rank Hankel operators, cf. 1.3 of [30] for more details. The maximum of |T 1 | on the circle is (cos η) −1/2 (sin(η)) 2 1 as η → 0. It follows that for any q > 0. Similarly for any q > 0 and we have proved u 2 is very good. Note that  Proof. It is sufficient to prove that (z −2 −u −2 )χ I 2 and its complex conjugate (z 2 −u 2 )χ I 2 are very good. The proof for (z −2 − u −2 )χ −I 2 and its complex conjugate is similar. To simplify writing we will denote the function (z −2 − u −2 )χ I 2 by h 1 only in the proof of this proposition. We first show that |(1 − P)h 1 P| S q = O(1) for some 0 ≤ q < 1. By Lemma 2.3 it is enough to show that I 2 h 1 z n dz = O(n −2 ), ∀n ≥ 0. On N we shall need an estimate of derivatives of h 1 similar to that of formula (22) for h on the circle. Note that when η is small enough, |z| is close to 1 on N . Since |z 2 − e −iη cos η| = sin η on N , cos η ∼ 1 − 1 2 η 2 , cos η − (cos η) −1 ∼ η 2 , we have |z 2 e iη − (cos η) −1 | ∼ η when η is sufficiently small, and hence sin η | is close to 1 when η is sufficiently small. Now the comments before formula (22) applies verbatim, We have both h 1 and h 1 are O(1) and |h 1 | ∼ 1/η, and it follows that has poles inside the unit disk, so our deformation of path argument above does not work for h * . But h * 1 = −z 2 u 2 h 1 , hence if we multiply h * 1 by 1 −z 2 u 2 then we get h 1 , and we have removed the poles of h * 1 . The function 1 −z 2 u 2 has modulus 1 and both 1 −z 2 u 2 and its complex conjugate are very good by Prop. 3.4, it will follow that h * 1 is very good. This will be called as "the trick of removing poles ". In more explicit terms, (Ph 1 (1 − P)) * = (1 − P)(−z 2 u 2 )h 1 P, by Prop. 3.2, Prop. 3.4 and |(1 − P)h 1 P| S q = O(1), 2/3 < q < 1 that has already been proved. Note that |Ph * 1 (1 − P)| S q = |(1 − P)h 1 P| S q and similarly with P replaced by 1 − P, and the proposition is proved.

Deformation of path:
The case with Branch cut. Our goal in this section is to show that h is good. Note that h is independent of the choice of branch cut of u inside unit disk , and in this section we choose the branch cut to be the closed line segment with end points e −iη/2 √ cos η and −e −iη/2 √ cos η. Here η is very close to 0 so that cos η ∼ 1.
Note that h has poles inside the unit disk, but since by Prop. 3.4 u −2 and u 2 are very good, by Prop. 3.2 it is sufficient to show that h 0 := u −2 h is good, and u −2 h has no poles in the unit disk, but has branch cut. This is another example of the trick of removing poles. The branch cut is important, because without the branch cut we c could conclude as in the previous section that by using the trick of removing poles, both g, h are very good, but this would contradict the lower bound in Lemma 2.4. First we have : where C is a constant with |C| = 1. Hence it is enough to check that is good. Note that z 1 = e iη z 2 and we think of h 1 as a function of z. Note that Exactly the same argument also shows that h 1 restricted to −I 2 verifies similar inequality.
Denote by a n = I 2 h 1 z n dz, n ≥ 0. Consider the function h 2 (z) = n≥0 a n z −n−1 on the circle. We will write h 2 as a sum of three functions. Note that h 1 z n is analytic in the unit disk except along the branch cut. We will write the I 2 h 1 z n dz as the integral of h 1 z n on three paths on the z plane. To describe these paths, note that we will be doing integrals in a small neighborhood of 1 when η is close to 0. In this small neighborhood the map z → z 1 = e iη z 2 is certainly one to one. Hence it is enough to describe these paths under the map z → z 1 = e iη z 2 .
The imagine of these three paths are easier to describe in terms of z 1 = e iη z 2 on the z 1 plane: first the path on the upper half of z 1 plane with |z 1 − cos η| = sin η from e iη to cos η − sin η; We denote this quarter of the circle byĴ 1 .
The second path is along part of the branch cut [cos η − sin η, cos η], and then turning in the opposite direction along the same closed interval. We denote this interval byĴ 2 . The last part is in the lower half of z 1 plane from cos η − sin η to e −iη , and we denote this quarter of the circle byĴ 3 . See Figure 2 for the image of the three paths on the z 1 plane. In Figure 2 points 2, 3 correspond to cos η − sin η, cos η respectively on the z 1 plane. The small arc part of the unit circle from e −iη to e iη is the image of I 2 on the z 1 plane. We will denote by J 1 , J 2 , J 3 the pre-images ofĴ 1 ,Ĵ 2 ,Ĵ 3 on the z plane.

The part from Integral Along a Quarter of a Circle Let us first show that
Recall that

Fig. 2. Image of a contour
When z 1 = cos η − sin η and η is sufficiently close to 0, we have 2 η, we conclude that the value of h 1 z n at z 1 = z 2 e iη = cos η − sin η is bounded by an absolute constant multiplied by We need the following: and the Lemma is proved.
By using integration by parts, h 1 is equal to 0 at z 1 = e iη , and Lemma 3.6 we have On J 1 we shall need an estimate of derivatives of h 1 similar to that of formula (22) for h on the circle. Note that when η is small enough, |z| is close to 1 on J 1 . Since |z 1 − cos η| = sin η on J 1 , |z 1 − (cos η) −1 | ∼ η when η is sufficiently small, and hence |m 1 (z 2 )| = | z 1 cos η−1 z 1 −cos η | is close to 1 when η is sufficiently small. Now the comments before formula (22) applies verbatim, and we find that h 1 z n is O(1) on the boundary of J 2 , and we have shown that Similarly we have

The part from Integrals Along the Branch cut
Set c n := J 2 h 1 z n dz. Since u changes signs from the upper part of the branch cut to the lower part, we should actually consider 2c n . But since our estimate is up to multiplication by a positive constant, we can ignore this constant 2 in the following. We need to show h 3 (z) := n≥0 c n z −n−1 is good. Since |(1 − P)h 3 P| q S q = |Ph * 3 (1 − P)| q S q , We will use inequality (24) for h * 3 . Letf n (t) = F n * h * 3 , and f n (t) =f n * . Then where the second equality follows since f n (t) is a function of t with period 1. We need to estimate −1/2≤t 1 ≤1/2 | f n (t)| p dt. On J 2 , and cos η cos η−sin η When η is sufficiently small we have 1 2z 1 ≤ 1. Note that j≥1 [ √ z 1 e −iη/2 ] j F( j/n)e −2πi jt = j≥1 F( j/n)e 2πi jt witht = t 1 + s 2π , s = −i 1 2 ln z 1 , and on J 2 , 1 2 ln z 1 < 0. When −1/2 ≤ t 1 ≤ 1/2 , apply Lemma 2.5 to j≥1 F( j/n)e 2πi jt , we have that up to O(n −N ) term for any N > 0, we can replace f n (t) by Now it is sufficient to evaluate ∞ m=0,n=2 m n |t 1 |≤ 1 2 |g n (t 1 )| p dt 1 . Note that by inequality (25) |F F(−n(t 1 + s 2π ))| ≤ C N e n 2 ln(z 1 ) 1 + n N (|t 1 + s 2π |)) N , ∀N ≥ 0 where the constant C N depends on N and F.
It follows that when η is sufficiently small We note that the exponential decay factor e n 2 ln(cos η) is due to the fact that the branch cut is inside the unit disk.
Recall that on J 2 , and cos η cos η−sin η It follows that |g n (t 1 )| ≤ C N n(− ln η)η 2 e n 2 ln(cos η) To evaluate −1/2≤t 1 ≤1/2 |g n (t 1 )| p dt 1 , t 1 = t − η 4π , we break this integral into two parts. Set δ := − ln(cos η). First we evaluate Note that η 2 = O(δ) and by Lemma 3.10 we have proved that Next we evaluate 1 2 ≥|t 1 |≥δ |g n (t 1 )| p dt 1 . This time we choose N in (26) such that 1 + p > N p > 1. Note that when η is small enough we have By inequality (24) we have proved Putting together these three parts from Sects. 3.3.1 and 3.3.2, and use Lemma 2.3, we prove the following Theorem:  It follows that and the Lemma follows.

Estimation of Entropy.
Since f 0 (Pg P), P 0 f (g)P are not trace class operators, but f 0 (Pg P)− P f 0 (g)P is, this makes it very delicate to show that S(F, η)−tr( f 0 (Pg P)− P f 0 (g)P) is small. This is proved in several steps in this section. We begin this section with a generalization of Lemma 2.2 in [38].
In our applications in this section 0 ≤ t 0 ≤ 2, 0 ≤ t 1 ≤ 1, 0 ≤ t 2 ≤ 1. So the maximum of t 0 (1 − σ ) + (t 1 + t 2 )σ ) is 2. There are two different cases that are important in the following: The first case is when we need t 0 (1 − σ ) + (t 1 + t 2 )σ < 2 to make sure our integral is convergent: in this case if we can manage to find one of the t i , i = 0, 1, 2 which do not take their maximal value then we will achieve our goal. The second case is when 0 ≤ t 0 ≤ 1, 0 ≤ t 1 +t 2 ≤ 1. In this case the maximum of t 0 (1−σ )+(t 1 +t 2 )σ ≤ 1, and we need to get t 0 (1 − σ ) + (t 1 + t 2 )σ < 1. Again this can be done if we can manage to get t 0 or t 1 + t 2 to take values less than their allowed maximum , then we can make sure that our integral is convergent. We will see three different such "savings" of the exponents in the following.
First we will use an integral representation for f 0 (T ) (cf. [7]).
Lemma 3.12. Suppose 0 ≤ T ≤ 1 is an operator. Then dβ and the Lemma follows from functional calculus for self-adjoint operators.
Let A = P P 12 P, A = P P 12 P(1 − P P 12 P). See Th. 2.1 for definition of P 12 .
Note that since P 12 is a projection, A = P P 12 P(1 − P P 12 P) = P P 12 Lemma 3.13.
So we have Apply Lemma 3.11 for W = 1 It follows that by Lemma 3.13 Recall that z = β 2 − 1/4, and so where S(F, η) is the first term in above integral by Lemma 3.12.
Next we estimate First we introduce some notations that will simplify writing. These notations will only be used in this section. Let . It follows from Th. 3.7, Cor. 3.9 and Prop. 3.2 that Recall by definition where W = h 2 1 . As a first step we estimate First we need a simple Lemma: Lemma 3.14. If S is a positive operator, and T T * ≤ S, z > 0, then Proof. Since ||Q|| = ||Q * ||, it is sufficient to prove We have Hence and the Lemma is proved.
Let us show that We will apply Lemma 3.11 with R 1 = 1 X +z , V = PW P − PW, R 2 = 1 W +z P. It is clear that t 1 = t 2 = 1, and we need choose t 0 small enough. The key observation is that and since X ≥ Ph 2 1 P, from Lemma 3.14 we have It is also clear that and we achieve our goal with Now we can apply Lemma 3.11 with t 0 = 3/2, t 1 = t 2 = 1 to obtain Since h 2 1 = −h 2 , By Th. 3.7 we get Let us consider We write Note that by Lemma 3.14 we have We have t 0 = 1/2, t 1 = t 2 = 1/2, again with savings on exponents. We have By Th. 3.7 and Cor. 3.9, the same argument as above shows that Finally we are left with , R 2 = (1 − P)g P 1 z+Y and use Lemma 3.14 we find that t 0 = 1/2, t 1 = t 2 = 1/2, again with savings as the preceding case to complete the proof that To summarize, we first prove that which is equation (27).
So we have proved the following theorem: where S(F, η) and τ (g, f 0 ) are defined as in definition 2.2.

Upper
Bound for Entropy. Th. 3.15 reduce the estimation of S(F, η) to τ (g; f 0 ). f 0 (Pg P) − P f 0 (g)P is called truncated Wiener-Hopf operators in [21]. There is a remarkable formula for τ (g; f 0 ) going back to H. Widom (cf. [21] and references therein). The more general version that we will use can be found in [37] and [21]. To describe this formula, we recall some basic definitions from [37]. Now we will use Cayley transformation to identify the unit circle with the extended real line, and to think our function g as a functionĝ on the real line, that isĝ(x) = g(C(x)). Recall Cayley transform V (x) = i(x + i)/(x − i), which carries the (one point compactification of the) real line onto the circle and the upper half plane onto the unit disk. It induces a unitary map of L 2 (S 1 , C) onto L 2 (R, C). The operator U carries the Hardy space on the circle onto the Hardy space on the real line (cf. Chapter one of [30]). We will use the Cayley transform to identify intervals on the circle with one point removed to intervals on the real line. Note thatĝ ∈ W 2,∞ , and has compact support. The length function is L(z) = |z 2 − e −iη cos η|. Notice that |L(z 1 ) − L(z 2 )| ≤ |z 2 1 − z 2 2 | ≤ 2|z 1 − z 2 |. Define scale function τ and amplitude function v ( cf. Sect. 3 of [21]) as follows: Note that One can check directly from (23) that where C, C k are constants. Note that the minimum of τ (x) is τ min = 1 − cos η ∼ η 2 . Ourĝ ∈ W 2,∞ , has support in [−2, 2] and verifies conditions 4.1 in [21]. Hence Th. 3.2 in [37] applies toĝ. where J is a path connecting end points of I 2 and −I 2 with |e iη z 2 − cos η| = sin(φ), since h(z)z 2n is analytic in the region bounded by I 2 ∪ −I 2 ∪ J ∪ −J . Here we have used the fact that h is independent of the choice of analytical continuation of u and we can choose branch cut of u which is outside the region bounded by I 2 ∪ −I 2 ∪ J ∪ −J . It follows that for as φ → 0, where C is a constant. It is also clear that is continuous in η. Since ih is real it follows all fourier coefficients of h goes to 0 when φ → 0. Similarly since g − 1 2 is odd we have )z 2n dz → 0 as φ → 0. Since on I 1 , g = 1, | I 1 z 2n dz| = O(φ). Moreover since g is real, it follows that all fourier coefficients of (g − 1 2 ) goes to 0 when φ → 0. By applying Lemma 3.19 (note that n = 0 in Lemma 3.19 ) and Prop. 3.20 with T 1 = P 12 (φ), T 2 = 0 we conclude that lim η→π − S(F, η) = 0. To prove continuity, we observe if we fix a small neighborhood V of η 0 in (0, π), then on V we have |h n | ≤ Cn −2 , |g n | ≤ Cn −2 where C only depends on the neighborhood V . Since h n , g n are obviously continuous in η, the continuity of S(F)(η) follows again by applying Lemma 3.19 and Prop. 3.20, with T 1 = P 12 (φ), T 2 = P 12 (η 0 ).